This theorem was inspired to me by this thread. I have found a proof of it but I'll post it later; I want to see if somebody can find another one!

Let $\displaystyle m(x), n(x)$ be continuous real-valued functions having periods $\displaystyle p,q$. Show that $\displaystyle m(x)+n(x)$ is periodic if and only if $\displaystyle p,q$ are linearly dependent over $\displaystyle \mathbb{Q}$.